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Knowledge of the world is often statistical.

• When the appearance of an object varies, but not completely arbitrarily.

• Examples: – Classes of objects: faces, cars, bicycles– Segmentation: contour shape, texture,

background appearance.

Represent statistics with probability distribution

• Every value has a probability

• Probabilities between 0 and 1

• Probabilities sum to 1

• Two Issues– How do we get these distributions?– How do we use them?

Background Subtraction

• We’ll use this as our first example.

• Many images of same scene.

• A pixel is foreground or background.

• Many training examples of background.

• Classify pixels in new image

The Problem

…

Look at each pixel individually

…Then classify:

Just Subtract?

-and threshold difference

10 80 120

Background isn’t static

1 2 3

4 10 100

Probability Distribution for Pixels

• p(I(x,y)=k) for the probability that the pixel at (x,y) will have an intensity of k

1,255

0

k

kyxIp

Bayes’ Law

)(

)()|()|(

DP

CPCDPDCP

kyxIP

yxBPyxBkyxIPkyxIyxBP

,

,,|,,|,

This tells us how to reach a conclusions using evidence, if we know the probability that the evidence would occur.

Probability (x,y) is background if intensity is 107? Who knows?

Probability intensity is 107 if background? We can measure.

)()|(or )()|(),( CPCDPDPDCPDCP

Bayes’ law cont’d

kyxIP

yxBPyxBkyxIPkyxIyxBP

,

,,|,,|,

yxFPyxFkyxIP

yxBPyxBkyxIP

kyxIyxFP

kyxIyxBP

,,|,

,,|,

,|,

,|,

If we have uniform prior for foreground pixel, then key is to find probability distribution for background.

Sample Distribution with Histogram

• Histogram: count # times each intensity appears.

• We estimate distribution from experience.• If 1/100 of the time, background pixel is 17,

then assume P(I(x,y)=17|B) = 1/100.• May not be true, but best estimate.• Requires Ergodicity, ie distribution doesn’t

change over time.

Sample Distribution Problems

• This estimate can be noisy.Try: k=6; n=10; figure(1); hist(floor(k*rand(1,n)), 0:(k-1))

for different values of k and n.

• Need a lot of data.

Histogram of One Pixel Intensities

Kernel Density Estimation

• Assume p(I(x,y)=k) similar for similar values of k.

• So observation of k tells us a new observation at or near k is more likely.

• Equivalent to smoothing distribution.

(smoothing reduces noise)

KDE vs. Sample Distribution

• Suppose we have one observation– Sample dist. says that event has prob. 1

• All other events have prob. 0

– KDE says there’s a smooth dist. with a peak at that event.

• Many observations just average what happens with one.

KDE cont’d

• To compute P(x,y)=k, for every sample we add something based on distance between sample and k.

• Let si be sample no. i of (x,y), N the number of samples, be a parameter.

N

i

iskN

kyxIP1

2

2

2)(exp

2

1,

KDE for Background Subtraction

• For each pixel, compute probability background would look like this.

• Then threshold.

Naïve Subtraction With Model of Background Distribution